Hypergeometric Functions and Feynman Diagrams

Kalmykov, M.Yu.; Kniehl, Bernd A.; Moch, S.; Ward, B. F. L.; Yost, S. A.

doi:10.1007/978-3-030-80219-6_9

Cited by 27 publications

(7 citation statements)

References 192 publications

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“… 26 The hypergeometric method was used to design a network with 1295 gene nodes and 1601 edges for commonly upregulated miRNA and 5954 gene nodes and 12 520 edges for commonly downregulated miRNA (Figure 4 ). 27 The target hit was 47 with adj.p value of 0.00009 for common upregulated miRNA of OSMF and OM samples, while the target hit was 185 with adj. p value of 7.87 e−19 for common down‐regulated miRNA of OSMF and OM samples.…”

Section: Resultsmentioning

confidence: 98%

Bioinformatics analysis of miRNA and its associated genes to identify potential biomarkers of oral submucous fibrosis and oral malignancy

Mohanan

Jhala

et al. 2023

Cancer Reports

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Background MicroRNAs are a group of non‐coding RNA that controls the gene expression. The interaction between miRNA and mRNA is thought to be dynamic. Oral cancer “The cancer of mouth” is quite prevailing in developing countries. miRNA has been found associated with oral cancer targeting tumor growth, cell proliferation, metastasis, invasion. The significant association of miRNA with genes could be used as a remarkable tool for diagnosis as well as prognostic analysis of oral cancer. Aim The aim of the present study is to evaluate common upregulated and downregulated miRNAs in oral submucous fibrosis (OSMF) and oral malignancy (OM) patients that can be used as diagnostic biomarkers, and to find out their interactions with target genes to establish associated networks in cancer pathways. Methods and Results Using miRDeep2 and DESeq analysis, the upregulated and downregulated miRNA in OSMF (Oral Submucous Fibrosis) and OM (Oral Malignancies) samples were compared to GEO (Gene Expression Omnibus) control dataset. There were 50 common downregulated miRNAs and 13 common upregulated miRNAs in OSMF and OM samples. miRNet analysis of common upregulated miRNA and common downregulated miRNA identified 1295 and 5954 genes, respectively connected with cancer pathways. From analysis of Hub genes, HRAS, STAT3, TP53, MYC, PTEN, CTNNB1, CCND1, JUN, VEGFA, KRAS were found associated with downregulated miRNA and VEGFA, TP53, MDM2, PTEN, MYC, ERBB2, CDKN1A, HSP90AA1, CCND1, AKTI were found associated with upregulated miRNA. The gene enrichment analysis of these hub genes were associated with cell communication, metabolic process, cell proliferation, and cellular component organization. Hub Genes linked with upregulated miRNA had an enrichment ratio of 11.828, whereas hub genes linked with downregulated miRNA had an enrichment ratio of 45.912. Conclusion We identified common deregulated miRNAs between OSMF and OM patients, which were further analyzed to find out associations with the genes correlated to cancer pathways. The hub genes identified in this study were found to have a significant impact on tumor growth and carcinogenesis. Also, the enrichment of these genes has revealed that the genes are associated with cellular communication, metabolic processes and various biological regulation. These deregulated miRNAs can be used to make a panel of biomarkers to diagnose oral cancer from blood even before its onset.

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Section: Resultsmentioning

confidence: 98%

Bioinformatics analysis of miRNA and its associated genes to identify potential biomarkers of oral submucous fibrosis and oral malignancy

Mohanan

Jhala

et al. 2023

Cancer Reports

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“…First, taking the residues of the Γ(−u l ) functions at negative integer points, the Mellin-Barnes integral (3) can be written as a linear combination of Horn-type hypergeometric series [10],…”

Section: Algorithmmentioning

confidence: 99%

Specializations of partial differential equations for Feynman integrals

Bytev,

Kniehl,

Veretin

2022

Preprint

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Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables z i , we derive a system of partial differential equations w.r.t. new variables x j , which parameterize the differentiable constraints z i = y i (x j ). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.

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“…The connection between Feynman integrals and the generalized hypergeometric functions was first proposed by Regge [64] and it was better understood with the knowledge of the Feynman integrals satisfying a holonomic differential equation [65], where the singularities of the differential equation were governed by the Landau singularities. The relation between Feynman integrals and hypergeometric system were studied in [66][67][68][69][70][71][72] (and reference therein). The isomorphism between the GKZ system and the Feynman integral was established in [63] and later realized within the LP representation in [73,74].…”

Section: Introductionmentioning

confidence: 99%

Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

Chestnov¹,

Gasparotto²,

Mandal³

et al. 2022

Preprint

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We elaborate on the connection between Gel'fand-Kapranov-Zelevinsky systems, de Rham theory for twisted cohomology groups, and Pfaffian equations for Feynman Integrals. We propose a novel, more efficient algorithm to compute Macaulay matrices, which are used to derive Pfaffian systems of differential equations. The Pfaffian matrices are then employed to obtain linear relations for A-hypergeometric (Euler) integrals and Feynman integrals, through recurrence relations and through projections by intersection numbers.

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Hypergeometric Functions and Feynman Diagrams

Cited by 27 publications

References 192 publications

Bioinformatics analysis of miRNA and its associated genes to identify potential biomarkers of oral submucous fibrosis and oral malignancy

Bioinformatics analysis of miRNA and its associated genes to identify potential biomarkers of oral submucous fibrosis and oral malignancy

Specializations of partial differential equations for Feynman integrals

Macaulay Matrix for Feynman Integrals: Linear Relations and Intersection Numbers

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